Dear FOM'ers
I find myself agreeing with Friedman rather than Forster on an NF
question; this is unusual...
The fact that NF and ZFC interpret arithmetic in different ways has
nothing to do with relative consistency strength issues. In NFU +
Infinity, arithmetic is interpreted just as in NF, but there is a
well-understood model theory (NFU is weaker than ZFC, and various
extensions of NFU are known, some weaker and some stronger than ZFC,
fitting into the linear hierarchy of consistency strength levels of
extensions (and fragments) of ZFC at reasonably well-understood
points.)
One could interpret arithmetic in ZFC in a way that looks more like
the NF approach (one could use Scott's trick). In any extension of
NFU or NF with Rosser's axiom of counting, one can use permutation
methods to ensure the finite von Neumann ordinals exist, and interpret arithmetic just as in ZFC.
What is known about NF on internal evidence is that it is at least as
strong as the theory of types with the axiom of infinity, or Zermelo
set theory with bounded comprehension. What is known on external
evidence about the strength of NF is ... nothing, since we have no
idea what a model would look like! My personal guess is that it is
probably consistent, with the lower bound just given as its actual
consistency strength, which would fit neatly in the linear hierarchy of consistency strengths, but I am open to any final outcome including inconsistency.
The reason that interpretation of arithmetic bears on consistency
strength has to do with the ability to code reasoning internally to
one's theory; this suggests that the details of the set theoretic
implementation of arithmetic can be expected to have very little to do
with consistency strength issues: it is simply a question of having
the usual operations of arithmetic available, which NF and ZFC (to all
appearances) do in the same way exactly. Now the ability to construct
sets of numbers, sets of sets of numbers, and so forth is highly
relevant to consistency strength: here all we can say is that NF can
do arithmetic of order n for each n, but apparently cannot do
arithmetic of order omega+1 or higher without additional assumptions;
advantage to ZFC, as far as we can tell...
--Randall Holmes